Problem: The lifespans of porcupines in a particular zoo are normally distributed. The average porcupine lives $16.4$ years; the standard deviation is $3.8$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a porcupine living between $12.6$ and $24$ years.
Solution: $16.4$ $12.6$ $20.2$ $8.8$ $24$ $5$ $27.8$ $95\%$ $68\%$ $13.5\%$ $13.5\%$ We know the lifespans are normally distributed with an average lifespan of $16.4$ years. We know the standard deviation is $3.8$ years, so one standard deviation below the mean is $12.6$ years and one standard deviation above the mean is $20.2$ years. Two standard deviations below the mean is $8.8$ years and two standard deviations above the mean is $24$ years. Three standard deviations below the mean is $5$ years and three standard deviations above the mean is $27.8$ years. We are interested in the probability of a porcupine living between $12.6$ and $24$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $95\%$ of the porcupines will have lifespans within 2 standard deviations of the average lifespan. It also tells us that $68\%$ of the porcupines will have lifespans within 1 standard deviation of the mean. The probability of a particular porcupine living between $12.6$ and $24$ years is ${68\%} + \color{orange}{13.5\%}$, or $81.5\%$.